The Amiga range of computers worked quite different, in that some rather peculiar colour modes were available. The original models with the OCS chipset could handle a palette of 4096 distinct colours. Depending on the resolution up to 32, 64 (halfbrite: 32 colours plus 32 half-shades of those colours), or even the full 4096 could be displayed at any time. There was never a 512 entry colour table, however.
Back on topic: it seems the OP wants more colours for the ANIMATED gif. Luckily, such an alternative animated web format exists: APNG (animate png). It is now supported in all browsers (even the new Edge!), and supports true colour and full alpha transparency (unlike animated gifs which are limited to 1bit transparency).
The issue is that none of the Adobe apps support animated PNGs, which is not surprising, because Adobe is notoriously slow in supporting newer file formats for the web (referring to the ridiculous reluctance to support webp). At this point we still need to rely on third-party tools to convert our animations to APNG (which generally means exporting the animation as single frames at full colour png files and importing these in an apng converter).
@conrad_C: there are tools which allow for a palette with entries going beyond the traditional 256: the brilliant ColorQuantizer (best visual PNG optimizer available right now) will support manageable indexed palettes with up to 4096 colours. http://x128.ho.ua/color-quantizer.html
Is there a reason it needs to be a GIF limited to that 256-color indexed color table? If it could be a JPEG or PNG, it could have 256 (non-indexed, continuous tone) colors per channel, for almost 17 million RGB colors. (Other formats that support indexed color, such as PNG and TIFF, are also limited to 256 colors.)
(Edit: After reading rayek_elfin’s informative post, I realized I had not noticed the GIF was animated. That changes my answer a little. Is your goal to make areas like the animated engine glow appear with less banding/posterization? Although you can’t use more than the 256 colors available in the GIF table, your other option is to apply dithering. It isn’t a perfect solution, it only trades off posterization for other artifacts, but sometimes careful amounts of dithering can be less distracting than banding.)
And I’m just guessing as to why you can’t go above 256, but my guess would be that since images are based on binary (1-bit, 4-bit, 8-bit…), the next step above above 8-bit would be 16 bits. Instead of 256 colors, 16 bits would get you over 65,000 colors. And that’s probably part of the problem: Nobody wants to manage a table that might have more than 65,000+ colors in it, it probably wouldn’t store colors efficiently as continuous tone, etc.